Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
A Priori Methods in Multiobjective Optimization Markus Hartikainen, PhD Department of Mathematical Information Technology, University of Jyväskylä, Finland
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Table of Contents 1
Recap on the Basics of Multiobjective Optimization
2
No Preference Methods
3
From No Preference Methods to a Priori Methods
4
General Properties of a Priori Methods
5
Examples of a Priori Methods
6
Augmentation Term
7
Critical Evaluation of a Priori Methods
8
From a Priori Methods to a Posteriori Methods Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Important Definitions in Multiobjective Optimization A general formulation for multiobjective optimization problems is min s.t.
f (x) := (f1 (x), . . . , fk (x))T x ∈ S,
where S ⊂ Rn is called the feasible set. A feasible solution x ∈ S is called Pareto optimal (PO), if there does not exist another solution y ∈ S such that 1 2
fi (y ) ≤ fi (x) for all i = 1, . . . , k and fj (y ) < fj (x) for some j ∈ {1, . . . , k }.
or weakly Pareto optimal (WPO), if there does not exist another solution y ∈ S such that fi (y ) < fi (x) for all i = 1, . . . , k . The set of objective function vectors given the Pareto optimal solutions is called the Pareto front (PF) Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
More Important Definitions in Multiobjective Optimization The ideal vector (or point) z ideal ∈ Rk has ziideal = min fi (x) for all i = 1, . . . , k . x∈S
The nadir vector (or point) z nadir ∈ Rk has zinadir =
max fi (x) for all i = 1, . . . , k .
x∈S is PO
A decision maker (DM) is a person that can give further preference information concerning the PO solutions
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Illustration of the Important Definitions in Multiobjective Optimization
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Different Methods for Multiobjective Optimization Often multiobjective optimization methods are classified w.r.t. to the role of a decision maker No preference methods are methods where the DM is not needed A priori methods are methods where the DM articulates preferences before optimization A posteriori methods aim to generate a representative set of PO solutions and the DM chooses the best one among them Interactive methods allow the DM to guide the search by alternating optimization and preference articulation iteratively
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
No Preference Methods Usually used only when the decision maker is not available e.g., online optimization Only one solution is computed E.g., the so-called method of global criterion where an optimization problem of the form min kf (x) − z ideal kp s.t. x ∈ S is solved, where k · kp is any Lp norm
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
From No Preference Methods to a Priori Methods In a priori methods 1 2
the preferences of the decision maker are asked and the best solution according to the given preferences is found
+ The DM can tell what kind of solution he wants - The DM has to devote some time to giving the preferences - The DM must understand the type of preference information - The DM must have some kind of (a priori) understanding about at least one of the following his own preferences the interdependencies of the objectives or the feasible objective values
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
A Priori Methods
A priori methods differ from each other in two main aspects: 1 2
How the preferences are asked? How (achieving) these preferences is modeled?
The preferences can be asked e.g., as reservation or aspiration levels, or as weights representing relative importance of the objectives Achieving the given aspiration levels can be modeled as e.g., minimizing distance between the vector containing the aspiration levels and the solution found
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Desirable Properties of a Priori Methods
1
There is an easily understandable way of getting the preference information from the DM
2
The DM’s preferences can be accurately modeled in the chosen way
3
For all possible preferences, the solution found is PO
4
Each PO solution can be found with some preference
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Desirable Properties of a Priori Methods
1
There is an easily understandable way of getting the preference information from the DM
2
The DM’s preferences can be accurately modeled in the chosen way
3
For all possible preferences, the solution found is PO
4
Each PO solution can be found with some preference
!!UNFORTUNATELY, THERE DOES NOT EXIST A METHOD THAT WOULD FULFIL ALL THE PROPERTIES!!
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
The -Constraint Method Means optimizing problem min s.t.
fi (x) x ∈ S, fj (x) ≤ j for all j ∈ {1, . . . , k } \ {i}.
(1)
Above, the j ∈ R are bounds for objectives and fi is the objective to be minimized, all chosen by the DM Objective bound j means that the DM wants a solution with fj (x) ≤ j
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Properties of the -Constraint Method For any ∈ Rk −1 , an optimal solution to (1) (if exists) is WPO For any ∈ Rk −1 , the unique optimal solution to (1) (if exists) is PO Let x ∈ S be a PO solution and let i ∈ {1, . . . , k }. Then by choosing j = fj (x) the solution x is an optimal solution to (1).
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
The Achievement Scalarizing Function A method developed by Wierzbicki in 1986, where one optimizes problem f (x)−ziasp min maxi=1,...,k zinad −z ideal (2) i i s.t. x ∈ S. Above, the z asp is a vector containing the aspiration levels of the decision maker Aspiration level ziasp means that decision maker would like to have a solution with fi (x) = ziasp
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Properties of the Achievement Scalarizing Function For any z asp ∈ Rk , an optimal solution to (2) is WPO For any z asp ∈ Rk , the unique optimal solution to (2) is PO Let x ∈ S be a PO solution. Then by choosing z asp = f (x) the solution x is an optimal solution to (2).
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
The Weighted Sum Means optimizing problem min s.t.
P
i=1,...,k
wi fi (x)
x ∈ S.
(3)
Above, the wi ≥ 0 are weights given by the decision maker representing the relative importance of the objectives If weight wi is bigger than wj it means that the decision maker appreciates improvement on objective fi more than on objective fj
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Properties of the Weighted Sum If wi ≥ 0 for all i = 1, . . . , k and wj > 0 for some j = 1, . . . , k , then an optimal solution to (3) is WPO If wi > 0 for all i = 1, . . . , k , then an optimal solution to (3) is PO However, there may be PO solutions that do not optimize (3) for any w ∈ Rk+ !
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Methods for Eliciting the Weights There exist multiple methods for eliciting the weights in weighted method Swing Weighting Lotteries and expected utilities etc.
Especially, in discrete choice situations these are used a lot However, no matter what method for eliciting the weights is used, it does not remove the problems of the weighting method In these lectures, we are not going to deal with these
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Augmentation Term Many a priori methods e.g., the above mentioned -constraint method and the Achievement scalarizing function guarantee only that (possibly not unique) solutions are merely WPO To guarantee PO solutions, so-called augmentation term X ρ fi (x) (where ρ > 0 is a small constant) i=1,...,k
is often added to the objective functions of a priori methods More often than not, augmented versions are used in practice instead of the original versions Also, other approaches for removing the WPO solutions exist Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Level Curves of the -Constraint Method with and without the Augmentation Term
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Common Problems in all a Priori Methods
1
The decision maker may not know what is feasible and what is not
2
The decision maker may not understand how the elicited preferences affect the solutions generated
3
Potentially better solutions may be missed as there is no feed back to the given preferences
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Also: the Weighted Sum May Produce Unintuitive Solutions
Assume choosing a wife from the following candidates: Martta Johanna Nina Relative importances (weights)
Appearance 1 5 10 0.4
Cooking 10 5 1 0.2
House keeping 10 5 1 0.2
Cleanliness 10 5 1 0.2
sum 6.4 5 4.6
Two big problems emerge: 1 2
The winner is the worst one in the most important criterion! Intuitively appealing compromise Johanna will not be chosen with any given weights!
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Also: the Weighted Sum May Produce Unintuitive Solutions
Assume choosing a wife from the following candidates: Martta Johanna Nina Relative importances (weights)
Appearance 1 5 10 0.4
Cooking 10 5 1 0.2
House keeping 10 5 1 0.2
Cleanliness 10 5 1 0.2
sum 6.4 5 4.6
Two big problems emerge: 1 2
The winner is the worst one in the most important criterion! Intuitively appealing compromise Johanna will not be chosen with any given weights!
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Also: the Weighted Sum May Produce Unintuitive Solutions
Assume choosing a wife from the following candidates: Martta Johanna Nina Relative importances (weights)
Appearance 1 5 10 0.4
Cooking 10 5 1 0.2
House keeping 10 5 1 0.2
Cleanliness 10 5 1 0.2
sum 6.4 5 4.6
Two big problems emerge: 1 2
The winner is the worst one in the most important criterion! Intuitively appealing compromise Johanna will not be chosen with any given weights!
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Also: the Weighted Sum May Produce Unintuitive Solutions
Assume choosing a wife from the following candidates: Martta Johanna Nina Relative importances (weights)
Appearance 1 5 10 0.4
Cooking 10 5 1 0.2
House keeping 10 5 1 0.2
Cleanliness 10 5 1 0.2
sum 6.4 5 4.6
Two big problems emerge: 1 2
The winner is the worst one in the most important criterion! Intuitively appealing compromise Johanna will not be chosen with any given weights!
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Also: the Weighted Sum May Produce Unintuitive Solutions
Assume choosing a wife from the following candidates: Martta Johanna Nina Relative importances (weights)
Appearance 1 5 10 0.4
Cooking 10 5 1 0.2
House keeping 10 5 1 0.2
Cleanliness 10 5 1 0.2
sum 6.4 5 4.6
Two big problems emerge: 1 2
The winner is the worst one in the most important criterion! Intuitively appealing compromise Johanna will not be chosen with any given weights!
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Also: the Weighted Sum May Produce Unintuitive Solutions
Assume choosing a wife from the following candidates: Martta Johanna Nina Relative importances (weights)
Appearance 1 5 10 0.4
Cooking 10 5 1 0.2
House keeping 10 5 1 0.2
Cleanliness 10 5 1 0.2
sum 6.4 5 4.6
Two big problems emerge: 1 2
The winner is the worst one in the most important criterion! Intuitively appealing compromise Johanna will not be chosen with any given weights!
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Also: the Weighted Sum May Produce Unintuitive Solutions
Assume choosing a wife from the following candidates: Martta Johanna Nina Relative importances (weights)
Appearance 1 5 10 0.4
Cooking 10 5 1 0.2
House keeping 10 5 1 0.2
Cleanliness 10 5 1 0.2
sum 6.4 5 4.6
Two big problems emerge: 1 2
The winner is the worst one in the most important criterion! Intuitively appealing compromise Johanna will not be chosen with any given weights!
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Also: the Weighted Sum May Produce Unintuitive Solutions
Assume choosing a wife from the following candidates: Martta Johanna Nina Relative importances (weights)
Appearance 1 5 10 0.4
Cooking 10 5 1 0.2
House keeping 10 5 1 0.2
Cleanliness 10 5 1 0.2
sum 6.4 5 4.6
Two big problems emerge: 1 2
The winner is the worst one in the most important criterion! Intuitively appealing compromise Johanna will not be chosen with any given weights!
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
Also: the Weighted Sum May Produce Unintuitive Solutions
Assume choosing a wife from the following candidates: Martta Johanna Nina Relative importances (weights)
Appearance 1 5 10 0.4
Cooking 10 5 1 0.2
House keeping 10 5 1 0.2
Cleanliness 10 5 1 0.2
sum 6.4 5 4.6
Two big problems emerge: 1 2
The winner is the worst one in the most important criterion! Intuitively appealing compromise Johanna will not be chosen with any given weights!
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
Recap on the Basics of Multiobjective Optimization No Preference Methods From No Preference Methods to a Priori Methods General Properties of a Priori Methods Examples of a Priori Methods Augmentation Term Critical Evaluation of a Priori Methods From a Priori Methods to a Posteriori Methods
From a Priori Methods to a Posteriori Methods
In a posteriori methods, a set of solutions is generated and the decision maker may choose the best one from that Pluses: They allow the decision maker to see what is feasible In theory, they guarantee that no superior solutions will be missed
Minuses: More computation is needed It may be hard for the decision maker to choose from a large list
Markus Hartikainen, PhD
A Priori Methods in Multiobjective Optimization
